Theory of Elasticity (Foundations of Engineering Mechanics)
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The classical theory of elasticity maintains a place of honour in the science ofthe behaviour ofsolids. Its basic definitions are general for all branches of this science, whilst the methods forstating and solving these problems serve as examples of its application. The theories of plasticity, creep, viscoelas ticity, and failure of solids do not adequately encompass the significance of the methods of the theory of elasticity for substantiating approaches for the calculation of stresses in structures and machines. These approaches constitute essential contributions in the sciences of material resistance and structural mechanics. The first two chapters form Part I of this book and are devoted to the basic definitions ofcontinuum mechanics; namely stress tensors (Chapter 1) and strain tensors (Chapter 2). The necessity to distinguish between initial and actual states in the nonlinear theory does not allow one to be content with considering a single strain measure. For this reason, it is expedient to introduce more rigorous tensors to describe the stress-strain state. These are considered in Section 1.3 for which the study of Sections 2.3-2.5 should precede. The mastering of the content of these sections can be postponed until the nonlinear theory is studied in Chapters 8 and 9.
=4h (t) +4h (t) +3, (t) + 4h (t) + 8h (t) , t; (gX) = 3- (5.4.5) 4h (£) + 4/2 (£) , (£) + 4/2 (£) - 8h (£) . (5.4.6) The ~ependences between the principal invariants of the strain tensors t t and are more complicated and can be obtained by means of formulae (5.2.4) . For example, 2.5 Relation between the strain measures 109 and due to eq. (5.4.5) (5.4.7) 2.5.5 Dilatation The elementary volumes of t he medium in the initial and act ual st at e are as follows dro = r 1 . (r2 x r3)
by determinant A = IAs,·1 of this matrix. Th en grt -_ /\\ sr /\\ st , "crt -_ ~2 (l:Urt and furthermore _ /\\ sr /\\ st ) (6.1.5) II Asr II (6.1.6) 114 2. Deformation of a cont inuum Let us notice th at bt i= 15t for the following reason. In the first case, bt denot es an extension of a linear element of unit length which was parallel to axis it in volume v , and extends to length 1 + bt in volume V. In the second case we speak about a linear element of length 1 + 15t in volume V which
expression for tensor E is also given by formulae (3.4.10) of Chapt er 3. The proof of the st ati onarity and minimality of th e functional in the equilibrium position does not differ from that in Subsection 4.2.5 provided that th e st at ically admissible states of stress are considered. 4.2.8 Saint- Venant 's principle. En ergetic consideration "T he principl e of th e elast ic equivalence of statically equivalent systems of forces" was first formulated for the problem of the state of st
area. Deliberately or not, an idealisation of the boundary conditions is always used for solving (correctly stated) problems of mathematical physics. In the problems of elasticity theory it is all the more unavoidable since the details of the distribution of the surface forces are most often unknown and the possibility of replacement by another distribution with the same integral properties seems to be intuitively acceptable. It is, however, clear that the above formulation of Saint- Venant's
volume v with a surface 0 reaches a new equilibrium state whose volume and surface are denoted by V and 0 respectively. The first state is referred to as the initial state (volume v or v-volume), whilst the second state is called the final state (volume V or V -volume) . In what follows , the concept of the natural state will be of importance. The natural state is the state when the continuum is not stressed. Unless otherwise stated, the natural state is not identified with an initial state. A